= Solution
For a <dominant integral weight> $\lambda$, let $V(\lambda)$ be the finite-dimensional irreducible <highest-weight representation> with <highest weight> $\lambda$. Set $\rho=\frac12\sum_{\alpha\in\Phi^+}\alpha$, the <half-sum of positive roots>. With $\alpha^\vee$ the <coroot> of $\alpha$, the <Weyl dimension formula> is
$$
\boxed{\dim V(\lambda)=\prod_{\alpha\in\Phi^+}
\frac{\langle\lambda+\rho,\alpha^\vee\rangle}
{\langle\rho,\alpha^\vee\rangle}.}
$$
Here $\Phi^+$ is the <positive system of a root system> specified by the chosen <root basis>. The <fundamental weights> are the basis dual to the simple <coroots>: $\langle\omega_i,\alpha_j^\vee\rangle=\delta_{ij}$. Thus $\lambda=\sum_i a_i\omega_i$ with nonnegative integers $a_i$. The denominators are positive; equivalently one can use $(\lambda+\rho,\alpha)/(\rho,\alpha)$ in each factor, since the coroot normalization cancels.
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